×

zbMATH — the first resource for mathematics

Drinfeld double of quantum groups, tilting modules, and \(\mathbb{Z}\)-modular data associated to complex reflection groups. (English) Zbl 07271141
Summary: Generalizing Lusztig’s work, Malle has associated to some imprimitive complex reflection group \(W\) a set of “unipotent characters”, which are in bijection of the usual unipotent characters of the associated finite reductive group if \(W\) is a Weyl group. He also obtained a partition of these characters into families and associated to each family a \(\mathbb{Z}\)-modular datum. We construct a categorification of some of these data, by studying the category of tilting modules of the Drinfeld double of the quantum enveloping algebra of the Borel of a simple complex Lie algebra. As an application, we obtain a proof of a conjecture by Cuntz at the decategorified level.
MSC:
20G42 Quantum groups (quantized function algebras) and their representations
18M15 Braided monoidal categories and ribbon categories
20F55 Reflection and Coxeter groups (group-theoretic aspects)
Software:
CHEVIE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. H. Andersen, Tensor products of quantized tilting modules,Comm. Math. Phys.,149 (1992), no. 1, 149-159.Zbl 0760.17004 MR 1182414 · Zbl 0760.17004
[2] B. Bakalov and A. Kirillov, Jr.,Lectures on tensor categories and modular functors, University Lecture Series, 21, American Mathematical Society, Providence, RI, 2001. Zbl 0965.18002 MR 1797619 · Zbl 0965.18002
[3] C. Bonnafé and R. Rouquier, An asymptotic cell category for cyclic groups,J. Algebra, 558(2020), 102-128.Zbl 07203048 MR 4102134 · Zbl 07203048
[4] N. Bourbaki,Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.Zbl 0186.33001 MR 240238 · Zbl 0186.33001
[5] M. Broué, G. Malle, and J. Michel, Towards spetses. I. Dedicated to the memory of Claude Chevalley,Transform. Groups,4(1999), no. 2-3, 157-218.Zbl 0972.20024 MR 1712862 · Zbl 0972.20024
[6] M. Broué, G. Malle, and J. Michel, Split spetses for primitive reflection groups. With an erratum to [MR 1712862],Astérisque, No. 359 (2014), vi+146pp.Zbl 1305.20049 MR 3221618 · Zbl 1305.20049
[7] A. Bruguières, Catégories prémodulaires, modularisations et invariants des variétés de dimension 3,Math. Ann.,316(2000), no. 2, 215-236.Zbl 0943.18004 MR 1741269 · Zbl 0943.18004
[8] V. Chari and A. Pressley.A guide to quantum groups, Cambridge University Press, Cambridge, 1994.Zbl 0839.17009 MR 1300632 · Zbl 0839.17009
[9] M. Cuntz, Fourier-Matrizen und Ringe mit Basis, Ph.D. thesis, Universität Kassel, 2005.
[10] M. Cuntz, Fusion algebras for imprimitive complex reflection groups,J. Algebra,311 (2007), no. 1, 251-267.Zbl 1163.20026 MR 2309887 · Zbl 1163.20026
[11] V. G. Drinfel’d, Quantum groups, inProceedings of the International Congress of Mathematicians. Vol. 1, 2 (Berkeley, Calif., 1986), 798-820, Amer. Math. Soc., Providence, RI, 1987.Zbl 0667.16003 MR 934283
[12] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik,Tensor categories, Mathematical Surveys and Monographs, 205, American Mathematical Society, Providence, RI, 2015. Zbl 1365.18001 MR 3242743 · Zbl 1365.18001
[13] P. Etingof and V. Ostrik, On semisimplification of tensor categories, 2018. arXiv:1801.04409
[14] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE—a system for computing and processing generic character tables. Computational methods in Lie theory (Essen, 1994),Appl. Algebra Engrg. Comm. Comput.,7(1996), no. 3, 175-210.Zbl 0847.20006 MR 1486215 · Zbl 0847.20006
[15] J. E. Humphreys,Introduction to Lie algebras and representation theory. Second printing, revised, Graduate Texts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1978. Zbl 0447.17001 MR 499562 · Zbl 0447.17001
[16] J. C. Jantzen,Lectures on quantum groups, Graduate Studies in Mathematics, 6, American Mathematical Society, Providence, RI, 1996.Zbl 0842.17012 MR 1359532 · Zbl 0842.17012
[17] A. Klimyk and K. Schmüdgen,Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.Zbl 0891.17010 MR 1492989 · Zbl 0891.17010
[18] A. Lacabanne, Slightly Degenerate Categories andZ-Modular Data,Int. Math. Res. Not. IMRN, Jun 2019.DOI: rnz105.
[19] G. Lusztig, Quantum groups at roots of1,Geom. Dedicata,35(1990), no. 1-3, 89-113. Zbl 0714.17013 MR 1066560 · Zbl 0714.17013
[20] G. Lusztig,Introduction to quantum groups. Reprint of the 1994 edition, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010.Zbl 1246.17018 MR 2759715 · Zbl 1246.17018
[21] G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen,J. Algebra,177 (1995), no. 3, 768-826.Zbl 0854.20057 MR 1358486 · Zbl 0854.20057
[22] G. Masbaum and H. Wenzl, Integral modular categories and integrality of quantum invariants at roots of unity of prime order,J. Reine Angew. Math.,505(1998), 209-235. Zbl 0919.57010 MR 1662260 · Zbl 0919.57010
[23] J. Michel, The development version of theCHEVIEpackage of GAP3,J. Algebra,435 (2015), 308-336.Zbl 1322.20002 MR 3343221
[24] M. Müger, Galois theory for braided tensor categories and the modular closure,Adv. Math., 150(2000), no. 2, 151-201.Zbl 0945.18006 MR 1749250 · Zbl 0945.18006
[25] M. Rosso, Analogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques,Ann. Sci. École Norm. Sup. (4),23(1990), no. 3, 445-467. Zbl 0721.17012 MR 1055444 · Zbl 0721.17012
[26] S. F. Sawin, Quantum groups at roots of unity and modularity,J. Knot Theory Ramifications,15(2006), no. 10, 1245-1277.Zbl 1117.17006 MR 2286123 · Zbl 1117.17006
[27] G. C. Shephard and J. A. Todd, Finite unitary reflection groups,Canad. J. Math.,6(1954), 274-304.Zbl 0055.14305 MR 59914 · Zbl 0055.14305
[28] T.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.